Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids
The collective excitations in water are studied based on the interactionsite model of liquids. Three collective modes, extracted from a generalized Langevin equation combined with the RISM theory of liquids, are identified as an acoustic mode and two optical modes. The drag force exerted on ions...
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Інститут фізики конденсованих систем НАН України
2001
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irk-123456789-1204252017-06-13T03:04:22Z Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids Hirata, F. Chong, S.-H. The collective excitations in water are studied based on the interactionsite model of liquids. Three collective modes, extracted from a generalized Langevin equation combined with the RISM theory of liquids, are identified as an acoustic mode and two optical modes. The drag force exerted on ions in water is described in terms of the response of these solvent collective excitations to the perturbation of ions. The ion-size dependence of the drag force, which has been a central issue in physical chemistry for long time, is studied in molecular detail based on the novel approach. 2001 Article Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids / F. Hirata, S.-H. Chong // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 261-276. — англ. 1607-324X PACS: 61.20.Gy, 61.20.Lc, 61.20.Qg, 61.25.Em DOI:10.5488/CMP.4.2.261 http://dspace.nbuv.gov.ua/handle/123456789/120425 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The collective excitations in water are studied based on the interactionsite
model of liquids. Three collective modes, extracted from a generalized
Langevin equation combined with the RISM theory of liquids, are identified
as an acoustic mode and two optical modes. The drag force exerted on ions
in water is described in terms of the response of these solvent collective
excitations to the perturbation of ions. The ion-size dependence of the drag
force, which has been a central issue in physical chemistry for long time, is
studied in molecular detail based on the novel approach. |
| format |
Article |
| author |
Hirata, F. Chong, S.-H. |
| spellingShingle |
Hirata, F. Chong, S.-H. Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids Condensed Matter Physics |
| author_facet |
Hirata, F. Chong, S.-H. |
| author_sort |
Hirata, F. |
| title |
Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids |
| title_short |
Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids |
| title_full |
Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids |
| title_fullStr |
Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids |
| title_full_unstemmed |
Collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids |
| title_sort |
collective density fluctuations and dynamics of ions in water studied by the interaction-site model of liquids |
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Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
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http://dspace.nbuv.gov.ua/handle/123456789/120425 |
| citation_txt |
Collective density fluctuations and
dynamics of ions in water studied by
the interaction-site model of liquids / F. Hirata, S.-H. Chong // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 261-276. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT hirataf collectivedensityfluctuationsanddynamicsofionsinwaterstudiedbytheinteractionsitemodelofliquids AT chongsh collectivedensityfluctuationsanddynamicsofionsinwaterstudiedbytheinteractionsitemodelofliquids |
| first_indexed |
2025-07-08T17:51:41Z |
| last_indexed |
2025-07-08T17:51:41Z |
| _version_ |
1837102115607019520 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 2(26), pp. 261–276
Collective density fluctuations and
dynamics of ions in water studied by
the interaction-site model of liquids
F.Hirata 1 , S.-H.Chong 2
1 Department of Theoretical Study, Institute for Molecular Science,
Okazaki, 444, Japan
2 Physik-Department, Technische Universität München,
85747 Garching, Germany
Received May 8, 2001
The collective excitations in water are studied based on the interaction-
site model of liquids. Three collective modes, extracted from a generalized
Langevin equation combined with the RISM theory of liquids, are identified
as an acoustic mode and two optical modes. The drag force exerted on ions
in water is described in terms of the response of these solvent collective
excitations to the perturbation of ions. The ion-size dependence of the drag
force, which has been a central issue in physical chemistry for long time, is
studied in molecular detail based on the novel approach.
Key words: molecular liquids, mode coupling, generalized Langevin
theory, water dynamics, ion dynamics
PACS: 61.20.Gy, 61.20.Lc, 61.20.Qg, 61.25.Em
1. Introduction
Study of dynamics in liquid water has scored considerable progress in the past few
decades. Especially, the molecular simulation method has made great contribution
revealing unique nature of the dynamics, which has its origin in the structural feature
characterized by the hydrogen-bond network.
Recently, two interesting papers have appeared, which concern collective exci-
tations in water. One of those is related to the acoustic mode. In general, if the
characteristic frequency ω of density fluctuations in liquid is plotted against the
wavenumber k, a linear relation is obtained in the small wavenumber limit, namely
limk→0 ω = ck, where c is the sound velocity. In the case of water, the sound ve-
locity is approximately 1500 m/sec. However, it has been disclosed by Teixeira and
coworkers based on the neutron scattering measurement that there is a ‘new’ sound
mode which propagates with velocity twice as fast as the ‘ordinary’ sound [1]. The
c© F.Hirata, S.-H.Chong 261
F.Hirata, S.-H.Chong
sound velocity is very close to that of ice. The existence of a similar sound mode
with a large velocity has been predicted by Rahman and Stillinger from the molec-
ular dynamics simulation [2]. These authors related the ‘new’ sound mode to the
dynamics of the hydrogen-bond network. On the other hand, such enhance in sound
velocity with increasing wavenumber is a phenomenon normally observed in liquids,
even in argon, which is known as the ‘positive visco-elastic effect’ [3]. Therefore, it
is hard to draw a definite conclusion at this moment with respect to which picture
is closer to the reality. An interesting question to be asked in this regard is how
the crossover occurs between the two sound modes when the wavenumber is varied.
More recently, an Italian group published an X-ray scattering result, showing that
the crossover takes place in continuous fashion [4]. It is not so easy for molecular
dynamics simulation to study such a problem, since there is a serious limitation con-
cerning the system size, or number of molecules to be simulated, which determines
the lower bound of the wavenumber. It would be more appropriate to rely on the
non-equilibrium statistical mechanics in this respect.
The other topic which has stimulated the field is collective excitations corre-
sponding to the “optical mode.” All other modes in which the characteristic fre-
quency ω does not vanish in the k → 0 limit are collectively referred to as optical
modes. Vallauri and coworkers extracted dispersion relations corresponding to the
optical modes from the analysis of the dynamic structure factor S(k, ω) of water [5].
The optical modes correspond apparently to the rotational motion of water, since the
model for water molecules is a rigid body and the vibrational excitation is entirely
prohibited.
2. Theory of liquid dynamics and collective excitations in w ater
The present study is based on the generalized Langevin equation which has
played a central role in the non-equilibrium statistical mechanics. “Generalization”
of the Langevin equation, often mentioned in literature, features a phenomenologi-
cal replacement of the friction and random forces by their counter parts accounting
for the non-Markovian nature of dynamics: for example, “colored noise” instead of
“white noise” for the random force. Such a description, however, does not meet the
requirements of chemistry, since chemical nature of environmental solvent around
a Brownian particle is too complicated to be explained by such phenomenological
parameters involved in the theory as the relaxation time of memory. There has been
a considerable effort devoted to the “generalization” of the Langevin theory based
on the projection-operator method, which relates the friction and random forces
with microscopic structure of liquids in terms of the density pair-correlation func-
tions [6]. Such methods enable us to treat liquid dynamics at a molecular level at
least for simple liquids consisting of spherical particles. However, the problem is not
so simple for molecular liquids which involve not only the translational motion but
also rotational motion of molecules. A straightforward solution to the problem is to
represent molecular center and orientation by six coordinates, and such a description
has been developed traditionally in literatures. The simplest example is the Debye-
262
Water and ion dynamics
Fick diffusion equation, which has been employed in the NMR relaxation theory.
The method, however, has obvious difficulty when it is applied to chemical process-
es. Firstly, using angular coordinates for rotational motions makes mathematical
formula too complicated even for small molecules. Secondly, such a treatment re-
quires an explicit description for coupling of rotational and translational degrees
of freedom, which is another nontrivial problem. Furthermore, it will be impossible
to apply the method to a chemical reaction in which molecules in concern change
their composition and/or geometry along a reaction path. The consideration stated
above made us construct a new model for dynamics of molecular liquids. What if
we regard dynamics of molecules as “translational” motions of constituent “atoms”,
which are correlated with each other? The Site-Site Smoluchowski-Vlasov (SSSV)
theory has been proposed based on such a consideration, in the most primitive level
of overdamped regime [7]. The model is a natural extension of the interaction-site
model in the equilibrium statistical mechanics to non-equilibrium process, and the
theory can be realized by combining the RISM theory with the generalized Langevin
theory. The model featuring a correlated translational motion of atoms has its own
disadvantage: the rotational motion of molecules is not explicitly described. The
problem has been solved by Chong and Hirata by extracting collective modes of the
density fluctuation from the site-site density correlation functions. In their recent
study for dynamics of a simple dipolar liquid based on the interaction-site model [8],
they have succeeded to abstract the collective excitations in liquids, which can be
identified as optical and acoustic modes, by diagonalizing the collective frequency
matrix appearing in the generalized Langevin equation. The two modes are related
essentially to the rotational and translational motions of molecules. In what follows,
we describe briefly the method applied to water.
Let us begin with the definition of the site-density fluctuations,
ρα(k, t) =
N
∑
i=1
exp[ik · rαi (t)], (1)
where r
α
i (t) denotes the position of α atom in ith molecule at time t, and N is the
total number of molecules. Here, α refers to oxygen atom (denoted as O) or one of
two hydrogen atoms (H1 and H2) constituting a water molecule. The site-site density
correlation function is defined by
Fαβ(k, t) =
1
N
〈ρα(k, 0)
∗ρβ(k, t)〉, (2)
the initial value of which is the site-site static structure factor, Fαβ(k, 0) = χαβ(k).
The generalized Langevin equation for Fαβ(k, t) is written as [8]
F̈(k, t) + 〈ω2
k〉F(k, t) +
∫ t
0
dτ K(k, t− τ)Ḟ(k, τ) = 0, (3)
where K(k, t) represents the memory function, and the matrix notation is used for
site-site correlation functions such as Fαβ(k, t). 〈ω
n
k〉 denotes the normalized nth
263
F.Hirata, S.-H.Chong
frequency-moment matrix of S(k, ω) defined by the relations,
S(k, ω) =
∫
∞
−∞
dt eiωt F(k, t), (4)
〈ωn
k〉 =
[
1
2π
∫
∞
−∞
dω ωn
S(k, ω)
]
χ
−1(k). (5)
If the damping term is absent, equation (3) is formally identical to that of the har-
monic oscillator, q̈+ω2q = 0. The eigen modes of the collective density fluctuations
can be obtained by diagonalizing the matrix 〈ω2
k〉, whose explicit expression for a
three-site water model can be found elsewhere [9]. The eigen frequencies obtained
by diagonalizing 〈ω2
k〉 for water are plotted against the wavenumber k in figure 1a.
There are three eigen frequencies, one of which vanishes in the k → 0 limit. By
definition, the mode can be identified as an ‘acoustic’ mode. The asymptotic form
of the eigen frequency of the mode becomes in the non-damping case [9] as follows:
ω2
acou(k → 0) =
kBT
Mχ(k = 0)
k2, (6)
where M is the total mass of a molecule, and we have noticed that all the site-
site static structure factors coincide in the k → 0 limit, denoted as χ(k = 0). The
ω2
acou(k → 0) coincides with that of the ordinary sound mode propagating with the
isothermal sound velocity (kBT/Mχ(k = 0))1/2 and the wavenumber k [10]. The fact
that the above expression includes the total mass of a molecule suggests the mode
is related to the translational motion of molecules.
The other two eigen frequencies, on the other hand, do not vanish in the k → 0
limit. By an analogy to the solid state physics, the other modes can be identified
as ‘optical’ ones. These modes are concerned with the relative motions of atoms,
whose characteristic frequencies do not vanish even in the k → 0 limit. The limiting
behaviours of the two optical modes, to be called OM-I (the optical mode I) and
OM-II (the optical mode II), are respectively given by [9]
ω2
OM−I(k → 0) =
2kBT
3χ′′(k = 0)
(zH − zO)
2 (1/Ix + 1/Iy), (7)
ω2
OM−II(k → 0) = kBT (1/Ix + 1/Iz), (8)
where Ix, Iy and Iz are the moments of inertia around the principal x-, y- and
z-axes, and zO and zH are the z-coordinates of O and H atoms of a molecule in
the body-fixed coordinate (see figure 1 of [9]). The optical modes are related to
the rotational motion of molecules as can be inferred from the appearance of the
moments of inertia in the above expressions. OM-I has a the second moment of the
density correlation function, χ′′(k = 0), in its expression, which is closely related to
the dielectric constant of liquid [11,12]. It also indicates that OM-I has collective
character. On the other hand, OM-II is essentially a single-molecule mode since the
expression is not associated with any collective density correlation function.
264
Water and ion dynamics
0.0 2.0 3.0 4.0 5.01.0
k [Å ]-1
160
120
80
40
ω
[p
s
]
-1
0
(a)
0.0 2.0 3.0 4.0 5.01.0
k [Å ]-1
0.0
-1.0
1.0
0.0
-1.0
1.0
0.0
-1.0
1.0 (b)
(c)
(d)
X O
X O
X O
X H1
X H2
X , XH1 H2
X , XH1 H2
Figure 1. (a) Eigen frequencies as evalu-
ated by diagonalizing 〈ω2
k〉. Solid, dashed
and dash-dotted lines give the eigen fre-
quencies of the acoustic mode, OM-I
and OM-II, respectively. (b) xO (solid
line), xH1
(dashed line) and xH2
(dash-
dotted line) defined in the text corre-
sponding to the acoustic mode. Notice
that xH1
= xH2
holds in the whole k-
region. (c) xO (solid line), xH1
(dashed
line) and xH2
(dash-dotted line) corre-
sponding to OM-I. Notice that xH1
=
xH2
holds in the whole k-region. (d) xO
(solid line), xH1
(dashed line) and xH2
(dash-dotted line) corresponding to OM-
II. Notice that xH1
= −xH2
holds in
the whole k-region. From (b) to (d), xO,
xH1
and xH2
are normalized such that
x2O + x2H1
+ x2H2
= 1.
The contributions from each atom to
the mode can be extracted in the follow-
ing way. Diagonalizing the matrix 〈ω2
k〉
corresponds to turning the description
of the system in terms of the density
fluctuations ρα(k) to the one in terms
of their linear combination:
xO(k) ρO(k) + xH1
(k) ρH1
(k)
+ xH2
(k) ρH2
(k), (9)
where xO(k), xH1
(k) and xH2
(k) are the
components of the eigen vector corre-
sponding to the mode. Thus, by analyz-
ing the sign and magnitude of xα(k)’s, it
is possible to obtain the information on
how each atom contributes to the mode.
Plotted in figure 1b to 1d are the
contributions of each atom to the acous-
tic and two optical modes, respectively.
It is seen from figure 1b for the acous-
tic mode that xO(k) ≈ xH1
(k) = xH2
(k)
holds well in the small-k region, which is
consistent with the fact that the sound
mode stems from the center-of-mass
translational motion of the molecules,
i.e., each atom in the molecule equally
contributes to this mode. On the oth-
er hand, as can be seen from figure 1c,
OM-I is governed by the lighter hydro-
gen atoms over the entire wavelength
range, because the rotational motion of
a molecule is dominated by the motion
of the lighter atoms which are located
further from the center of mass. Fig-
ure 1d shows that OM-II is just relat-
ed to the rotational motion in which
two hydrogen atoms move in the out-of-
phase fashion with oxygen atom fixed.
More detailed analysis of the eigen vec-
tors revealed that OM-I and OM-II are
associated with the pitch and roll libra-
tional motions of the molecules, respec-
tively (see figure 4 of [9]). Since OM-II
turns out not to contribute to the ion
265
F.Hirata, S.-H.Chong
dynamics to be described in the next section, it shall not be considered any more in
the present article.
These resonances determined by 〈ω2
k〉 are shifted and damped by the memory
functions Kαβ(k, t). An approximation scheme has been developed for Kαβ(k, t)
which enables one to calculate the site-site density correlation functions Fαβ(k, t)
and their spectra [8]. This comprises the assumption of the exponential form for the
memory functions and the extension of Lovesey’s method for determining relaxation
times appearing in the assumed form for the memory functions. Figures 2 and 3 show
the results based on this theory along with the molecular-dynamics (MD) simulation
data. The results are reported in the form of longitudinal-current spectra, which give
the spectra of the collective excitations in liquids, defined by
CL,αβ(k, ω) =
ω2
k2
∫
∞
−∞
dt eiωt Fαβ(k, t). (10)
Furthermore, linear combinations of the form
CL,XX(k, ω) =
∑
α,β
cα cβ CL,αβ(k, ω), X = M or Z, (11)
rather than CL,αβ(k, ω) themselves are plotted in the figures since one can separately
discuss the acoustic and optical modes based on these combinations [13]. Here,
CL,MM(k, ω) and CL,ZZ(k, ω) denote the longitudinal-current spectra of the mass
and charge, respectively, which can be obtained by setting cα = mα (mass of atom
α) or cα = qα (partial charge of atom α) in equation (11).
It is seen from figure 2 that the theoretical results for the acoustic-mode spec-
tra are in fair agreement with MD simulation results; especially, the peak positions
of CL,MM(k, ω) are well reproduced. (The discernible lower frequency peak in MD
simulation result for n = 4 in figure 2 can be ascribed to a single-molecule excita-
tion, and not to the collective acoustic one, as exhibited by Miura [14].) Concerning
the width of the spectra, the agreement is not so satisfactory, and the use of the
recently developed mode-coupling theory for molecular liquids [15,16] may improve
the results in this regard. Compared in figure 3 are the theoretical and MD simula-
tion results for the longitudinal current spectra of the optical dynamics, CL,ZZ(k, ω).
As can be seen from figure 3, the theoretical results for the peak frequencies are
considerably lower than those of MD simulation. However, the overall shape of the
spectra is well reproduced by our theory as can be checked by shifting the theo-
retical results horizontally such that their peak positions coincide with those of the
simulation results. It was discussed that the disagreement in the peak positions of
CL,ZZ(k, ω) is largely due to the disagreement between the theoretical and simulation
results for the fourth frequency moment 〈ω4
k〉 [13]. Thus, although some quantitative
disagreements can be seen between the theoretical and MD simulation results, the
qualitative nature of the collective density excitations in liquid water is well cap-
tured by the present theory. Some other features on collective excitations in water,
such as the high-frequency sound velocity and the dispersion relations, have also
been discussed in [9,13], showing that all the essential features, reported previously
266
Water and ion dynamics
ω [ps ]-1
0 100 2000 100
0.0
0.5
1.0
0.5
1.0
C
(k
,ω
)
L n = 1
n = 2
n = 3
n = 4
Figure 2. The theoretical (solid lines) and MD simulation (circles) results of
the longitudinal current spectra for the acoustic dynamics, CL,MM(k, ω). The
results are plotted as a function of ω for k = n kmin with kmin = 0.3185 Å−1 and
n = 1 ∼ 4, in arbitrary units. kmin refers to the minimum accessible wave vector
from MD simulation.
ω [ps ]-1
0 100 2000 100
0.0
0.5
1.0
0.5
1.0
C
(k
,ω
)
L n = 1
n = 2
n = 3
n = 4
Figure 3. The theoretical (solid lines) and MD simulation (circles) results of
the longitudinal current spectra for the optical dynamics, CL,ZZ(k, ω). The other
notation is the same as in figure 2.
267
F.Hirata, S.-H.Chong
by neutron-scattering experiments [1], MD simulations [5,14,17–19] and dielectric
theories [20,21] are well reproduced.
3. Dynamics of solvated ions in water
Dynamics of ions in polar liquids has attracted scientists’ attention for a long
time in many different fields including physics, chemistry and biology. Main interests
have been focused upon the puzzling behaviour exhibited in the ion-size dependence
of the friction coefficient, which decreases with increasing ion-radii as opposed to
the prediction of the hydrodynamic Stokes law. Two classical models which are
based on entirely different physics have been practiced to explain the paradoxical
behaviour. The first of those is the so-called ‘solvent-berg’ model in which an ion
makes a ‘complex’ with solvent molecules, and undergoes Brownian motion with
the ‘effective Stokes radius’ of the molecular complex. The effective radius decreases
with increasing radius of a bare ion, so does the Stokes friction, because the elec-
trostatic attraction between ion and solvent decreases with increasing ionic radius.
The dependence of the ion mobility on its size has also been explained from an
entirely different point of view in terms of the dielectric friction. When an ion is
placed in the polar medium, surrounding solvent is polarized due to the electric field
of the ion. If the ion is displaced, the relaxation of the solvent polarization should
take place to accommodate itself to the new position of the ion. The relaxation
process is associated by an energy dissipation, and the corresponding drag force can
be identified as the dielectric friction. The dielectric friction reduces with decreas-
ing electrostatic field as the ion size increases. The two models, solvent-berg with
the effective Stokes radii and the dielectric friction, explain at the same qualitative
level the observed feature of the ion-size dependence of dynamics of small ions in
polar liquids; the friction reduces with increasing ionic radii when the ions are suf-
ficiently small, and then it begins to enhance as the radii further increase. As long
as the general behaviour is concerned, all the theoretical treatments proposed so
far, pure hydrodynamic [22–24], composite hydrodynamic and microscopic [25,26],
could apparently have explained a qualitative nature of ion dynamics projected on
the friction coefficient. However, the theories have left many important questions
unsolved. Which of the above two models for ion dynamics is more realistic? If both
processes are taking place, how are they coupled? What is the qualitative difference
which distinguishes a solvent from other solvents with respect to ion dynamics?
Additional questions are raised when the moving ion is immersed in water in-
stead of a simple dipolar liquid. The hydration phenomena observed by many dif-
ferent experiments show a remarkable variety in its dependence on size and sign
of ions, which may not be characterized by a simple chemical model such as ion-
water complex formation, or a solvent-berg model. Such a model seems to apply
to very small monovalent ions like Li+ and F− and multi-valent ions which make a
stable hydration shell with a substantial lifetime. However, water molecules in the
first hydration shell around ions with greater size are more mobile and disordered
than those in bulk water. The behaviour which has been referred to as ‘structure
268
Water and ion dynamics
breaking’ by Frank and Wen [27] and as ‘negative hydration’ by Samoilov [28] is due
to the competition between two forces acting on water molecules in the shell, the
isotropic electrostatic field from the ion and the hydrogen-bonding with other water
molecules. Such characteristics of ion hydration is expected to produce non-linearity
in the response of the electrostatic potential fluctuations by solvent to the ion field,
which has strong dependence on size of ions [29,30]. Recently, such hydration phe-
nomena have been characterized theoretically based on the RISM theory in terms of
the ion perturbation to the pair-correlation functions [31]. The following questions
may be naturally raised in conjunction of the ion dynamics in water. Does the dy-
namics of ions in water reflect such characteristics of the equilibrium structure of
hydration? If it does, how and to what extent? Is there any qualitative difference in
dynamics of ions in water compared to that in simple polar solvents, and what is
the difference if any? In this section, we try to answer some of these questions.
Here, we address the problem from the response of collective density fluctuations
in solvent, described in the previous section, to the ionic field [32,33]. Since the Stokes
and dielectric frictions originate basically from the energy dissipation associated with
the translational and rotational motions of solvent, respectively, it is reasonable to
ask how the ionic field couples to the collective density fluctuations, and/or how the
two drag forces are related to the two collective modes. Since the translational and
rotational motions of molecules are inherently coupled with each other in our atom-
based description of solvent dynamics, the theory is free from artifacts associated
with the decoupling of those motions, which is inevitable in the earlier theories based
on the explicit orientational coordinates.
3.1. Memory-function formulation of ion dynamics
Exploiting the standard fluctuation-dissipation theorem of Einstein, the friction
coefficient ζ is related to the diffusion coefficient D as
ζ = kBT/D. (12)
The diffusion coefficient in turn is related to the velocity auto-correlation function
Z(t) by the Green-Kubo formula [34]
D =
∫
∞
0
dt Z(t). (13)
The velocity auto-correlation function for an atomic ion in fluid is defined by
Z(t) = 〈vu,z(0) vu,z(t)〉, (14)
where vu,z(t) denotes the z-components of the velocity of the ion at time t. (The
subscript u signifies the solute ion.) The procedure of the projection-operator for-
mulation leads to the memory-function equation for Z(t) [34]:
Ż(t) = −
∫ t
0
dτ K(t)Z(t− τ), (15)
269
F.Hirata, S.-H.Chong
where K(t) denotes the memory function.
Following the procedure proposed by Sjögren [35,36], we approximate the mem-
ory function as
K(t) = Kfast(t) +Kslow(t), (16)
where Kfast(t) denotes the rapidly decaying portion of the memory function due to
the binary collisions, while Kslow(t) represents the slow portion arising from corre-
lated collisions. The fast portion can be well represented by the Gaussian ansatz,
Kfast(t) ≈ K(0) exp(−t2/τ 2), (17)
with the decay constant defined by
1/τ 2 = −
1
2
K̈(0)
K(0)
. (18)
In the expression,K(0) is the so-called “Einstein frequency”, which can be calculated
exactly from the information of the site-site intermolecular potential as well as of
the density pair-correlation functions. K̈(0) is also an equilibrium quantity but with
the three-particle correlation functions [32].
For the slow part of the memory function, we employ the mode-coupling theory,
which leads to [32]
Kslow(t) =
ρkBT
6π2m
∫
∞
0
dk
∑
λ,µ
cuλ(k)cuµ(k)(1− fu(k, t))Fu(k, t)Fλµ(k, t), (19)
where m is the mass of an ion, cuλ(k) denotes the site-site direct correlation function
between the ion and solvent, and Fu(k, t) represents the self-intermediate scattering
function of the solute. We adopt the Gaussian approximation for Fu(k, t),
Fu(k, t) = exp
(
−k2
∫ t
0
dτ (t− τ)Z(τ)
)
, (20)
which is exact in the short- and long-time regime. Fλµ(k, t) is the solvent site-site
intermediate scattering function defined in the preceding section. fu(k, t) is an auxil-
iary function defined by F id
u (k, t)/Fu(k, t), where F
id
u (k, t) = exp[−(kBT/m)k2t2] de-
notes the intermediate scattering function of ideal gas. The velocity auto-correlation
function Z(t) and its memory function K(t) can be obtained as a self-consistent so-
lution of the above equations.
3.2. Velocity auto-correlation functions
The velocity auto-correlation functions (VACF) of alkali and halide ions in water
obtained from the theory just described are plotted against time in figure 4. The
general behaviour of VACF is high-lighted by the pronounced oscillation seen in the
small ions, which disappears with increasing ion size. The oscillation is apparently
270
Water and ion dynamics
Li +
Na+
K +
Cs+
Rb+
F -
Cl-
Br -
I -
N
or
m
al
iz
ed
Z
(
t)
0.5
1.0
0.0
-0.5
0.5
1.0
0.0
t [ps]
0.0 0.2 0.40.1 0.50.3
Figure 4. The normalized velocity auto-
correlation functions of cations and an-
ions in water.
due to the vibrational motion of the
ions. Similar pictures have been ob-
tained by Rasaiah and co-workers based
on the molecular dynamics simula-
tion [37]. This suggests the existence of
a solvent cage around the ions, whose
life-time is long enough to support the
vibration. Remember, the ions which
have pronounced oscillation in VACF
are those classified as of ‘positive’ hy-
dration by Samoilov [28]. Therefore,
the dynamic picture is well in harmo-
ny with the static one in this respect.
The oscillation disappears in two physi-
cal causes; the size and mass. As the ion
size increase, the ion-solvent interaction,
whose essential nature is electrostatic,
decreases, which loosen the solvent cage.
Increasing mass makes the ion motion
more ‘inertial’ with longer characteris-
tic time. On the whole, the behaviour of
VACF obtained from the theory is con-
sistent with our intuitive picture for ion
dynamics in water.
3.3. Diffusion constants
The diffusion constants calculated by equation (13) using VACF described above
are depicted in figure 5 as functions of the ion radius, which is taken as half of the
Lennard-Jones σ parameter. The behaviour is striking in a sense that it entirely
breaks the Einstein-Stokes law, which predicts a monotonic decrease of the diffusion
constant as the ion size increases. (Though we have not shown here, our theory
in fact predicts the monotonic decrease, when the electrostatic interaction between
the ion and solvent molecules is ‘turned off.’) Instead, our results exhibit just the
opposite behaviour to the Einstein-Stokes law when the ion size is small, and turn
into the same trend as the ion size is further increased. The physical cause of the
anomalous behaviour seen in the diffusion constant we will discuss in detail lat-
er, because the diffusion constant contains essentially the same information as the
friction coefficient. Here, we would like to make just two points. By comparing the
plots for the cations (upper panel) and for the anions (lower panel), one can read-
ily find that those curves do not fall on top of each other. The asymmetry with
respect to the sign of charges comes from the asymmetry of charge distribution in
a water molecule. As has been addressed in many ways, structure of water is char-
acterized by the well-developed hydrogen-bond network, which has its origin in the
271
F.Hirata, S.-H.Chong
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